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A particle in one dimension moves under the influence of a potential $V(x)= ax^6$, where $a$ is a real constant. For large $n$, what is the form of the dependence of the energy $E$ on $n$?

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Hi Tanuj - is this in fact a homework question, or homework-like? (In other words, what's your motivation for asking this question?) We can use that to give you an answer that's most useful for you. –  David Z Aug 23 '11 at 1:57

2 Answers 2

For large $n$, the semiclassical approximation is valid and for bound states we may use the Bohr-Sommerfeld quantization condition:

$n = \frac{1}{h}\oint p dq$, where $n$ is the principal quantum number, $h$, the Planck constant, $q$, the position and $p$, the momentum on the classical trajectory.

In our case due to the conservation of energy:

$\frac{p^2}{2m}+ax^6 = E \rightarrow p = \sqrt{2m(E-ax^6)}$,

where $m$ is the mass and $E$, the total energy.

By substitution, we obtain:

$n = \frac{1}{h}\int_{-(E/a)^{1/6} }^{(E/a)^{1/6} }\sqrt{2m(E-ax^6)} dx$,

where the integration is between the two turning points. By scaling the integration variable:

$x =( \frac{E}{a})^{1/6}y$

We obtain:

$n = \frac{\sqrt{2m}}{h}{(\frac{E}{a})}^{2/3}\int_{-1 }^{1} \sqrt{(1-y^6)} dy$.

Thus:

$E \propto n^{3/2}$

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Shouldnt the $2m$ also be under the square root? Other wise good answer! –  Nic Aug 22 '11 at 15:09
    
Thanks @Nic, corrected –  David Bar Moshe Aug 22 '11 at 15:12
    
Great answer, but it looks like the "homework" tag has been added since you posted this, and although we can't be sure, it does sound like a homework question. Would you mind either editing out some of the details or temporarily deleting the answer until Tanuj confirms the nature of the question, so that we're not giving away a complete solution to somebody's homework? –  David Z Aug 23 '11 at 1:53
    
@ David Zaslavsky - I deleted the answer using the delete option at the bottom of the page, I hope that's the right way to do this –  David Bar Moshe Aug 23 '11 at 6:10
    
@David: yep, thanks. Since the question was asked a while ago it's probably safe to have full answers available now, so I undeleted yours. (Hope you don't mind) –  David Z Sep 4 '11 at 6:20

You might not be aware, but the quantum number "n" has a classical interpretation as the action variable "J". The action variable measures the area in phase space of the classical orbit,

$$J = \oint p {dx\over dt} dt$$

And the correspondence between J and n was known before quantum theory was developed. It is easy to work out the orbit shape in phase space for any value of J, and figure out what the dependence between E and J is, classically. This would solve your problem in the correspondence limit, in the limit of large n.

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